This is a question in the undergraduate-level textbook "Advanced Calculus" by Fitzpatrick.
Suppose that a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is twice differentiable such that for $\forall x$, $f'(x)\leq f(x)$, and $f(0)=0$. Then is $f$ the zero function?
The answer to this is not true as I was able to find a counter-example $f^*(x)= 1- e^x$. However we have only just learned about differentiation, the mean-value theorem and how to find extremes using 1st and 2nd derivatives, and we have only seen derivatives of polynomials so far, but I don't know how to disprove the above statement by using these.
(EDIT) For $1−e^x$ to be a valid counter-example, I need to "officially know" that the exponential function's derivative is equal to itself. But exponential functions are in the next chapter. Therefore unless I want to "cheat", I need to think of another function.
that are the closure of some open subset of $\mathbb{R}^2$, and
have area $x$.
For any $\Omega \in \mathcal{X}$ and any $x,y \in \Omega$, let $d(x,y\Omega)$ be the length of the shortest path lying entirely inside $\Omega$ and connecting $x$ to $y$.
Also, let $\mathbb{1}(x,y,\Omega)$ be the indicator function that equals $1$ if $x$ and $y$ are both in $\Omega$, and zero otherwise.
Is it possible to show that a solution to this problem must be convex?
Is the problem even well-defined (e.g., does $d(x,y,\Omega)$ always exist? I tried to help guaranteeing it by including only closed and connected sets in $\mathcal{X}$, but I am not 100% sure that's enough)?
Is the problem guaranteed to have a solution without more regularity conditions?
Some related problems are described here: https://link-springer-com.dist.lib.usu.edu/content/pdf/10.1007%2Fs10958-012-0717-3.pdf. Unfortunately, these related problems either (1) focus on "networks" rather than "thick" sets, (2) focus right away on convex sets, or (3) do not fix the area and instead impose an additional "cost" to the minimization problem for increasing the area.
I understand that differentiation of a function ($\mathbb{R} \rightarrow \mathbb{R} $) at a point is the rate of change in the output for a slight nudge in the input. And this rate of change could be negative or positive. There is no concept of direction for the single-variable function as obvious.
Now, my doubt is in the case of the multivariate function ($\mathbb{R}^n \rightarrow \mathbb{R}$) where differentiation is a gradient. And this gradient representing partial differentiation w.r.t. to each basis becomes a direction. This direction is a direction of ascent but not descent, why?. Why it is a direction is of ascent. My question is not at all related to steepest ascent, about which one can find many answers on this forum and read elaborately at this link. An intuitive explanation would be preferable than mathematical at this link.
I sometimes try to explain to my highschool students that those rules are not always true. For example, $0^{-2} \cdot 0^{2} = 0^0$ or I give other
interesting false deductions such as: $$\left(-1\right)^3=(-1)^{6\cdot \frac{1}{2}}=\left((-1)^{6}\right)^{\frac{1}{2}}=\sqrt{1}=1 $$
However I could not find an exact reference to where those rules are true.
Steward's Review of Algebra states that those rules are true if $a$ and $b$ are positive (real) numbers, and $n$ and $m$ are rational numbers. This is of course very conservative. Those rules are also true if $a\ne$, $b\ne 0$ and $n,m$ integers. Besides that I think many of those rules are also true if $n,m$ are real numbers.
So my question is, when are the above rules correct?
However, $x=\frac{1}{2}$ doesn't satisfy the original equation.
I understand that extraneous roots do creep in while solving inverse trig problems, but I wonder why it crept in here.
If possible (if it doesn't make the question too broad), I'd like to know the general causes for the occurrence of extraneous roots in inverse trig equations, too.
As the title says, I want to find a way calculate the following integral$ \int_{0}^{c} \frac{\sin(x/2)x~dx}{\sqrt{\cos(x) - \cos(c)}}$, which I know is equal to $\sqrt{2} \pi \ln(\sec(\frac{c}{2}))$.
At first glance, I thought this would not be very difficult to prove (and maybe it isn't), but after some straighforward manipulations, I was unable to make the $\pi$ factor appear.
Let's say I have $5$ unfair coins. Each with an independent, known, probability of landing on heads. I flip each coin once. How can I find the probability that I get $3$ or more heads?
Let $\gamma: (0,1) \to \mathbb{R}^2$ be a $C^\infty$ regular plane curve, and suppose that its curvature is at least $k_0$ everywhere. Is the image of $\gamma$ contained in disk of radius $\frac{1}{k_0}$?
Intuitively, it seems likely to be true. If we choose $p$ at a distance of $\frac{1}{k_0}$ from and perpendicular to $\gamma(0)$, then it seems like the curve will be inching towards $p$. However, I can't seem to prove that this is the case.
If the image of $\gamma$ is in $\mathbb{R}^3$ (or $\mathbb{R}^n$ for any $n > 2$), then I believe the result is false, since we can just take tightly wound helix that travels upwards very high. Therefore, if the result is true in $\mathbb{R}^2$, then I think it really uses the $2$-dimensional-ness of the space.
One thing I know is that if $|\gamma(t)|$ is maximized when $t = t_0$, then $|k(t_0)| \ge \frac{1}{|\gamma(t_0)|}$. Maybe we can translate the curve so that the $p$ defined above is the origin, and then use something like this somehow? The inequality goes in the wrong direction, so we probably need some sort of opposite result, but I'm not sure what that result would be.
I need help computing the value of the following definite improper integral:
$$\int_0^\infty \frac{x dx}{(1+x^2)(1+e^{\pi x})}=\text{?}$$
Here are my thoughts and attempts:
I tried using the Laplace Transform identity for definite integrals, with no luck (since I can only compute the Laplace Transform of $\frac{1}{e^{\pi x}+1}$ in terms of the digamma function... yuck)
I can't use the residue theorem, since the integral is from $0$ to $\infty$ and the integrand is not an even function
I would like to expand $\frac{x}{1+x^2}=\frac{1/x}{1-(-1/x^2)}$ as a geometric series, but it wouldn't always converge since $x$ goes from $0$ to $\infty$
CONTEXT:
The integral came up in Jack D'Aurizio's answer to this question.
Let me give you an example of what I mean. Flag algebras are a tool used in extremal graph theory which involve writing inequalities that look like:
(It's not too important to my question what this inequality means, but let me give you some context. Informally, the things we're adding and multiplying are probabilities that a random group of vertices in a large graph will induce some specific small subgraph. To make some manipulations rigorously justified, this is not precisely what we mean; instead, they are the limits of such probabilities over a convergent sequence of graphs.)
Aside from being potentially useful in solving math problems I'm curious about, I enjoy using, thinking about, and even looking at statements about flag algebras, because these equations and inequalities just look so cool! Instead of multiplying, adding, and comparing letters and numbers, we get to do the same thing to pictures of things.
So my question is: what are some other topics in mathematics where we get to do the same thing?
Obviously, you can always give any name you like to a variable, like those math problems you see on facebook where cherry plus banana is equal to three times hamburger. I'm not interested in examples like this, because there's nothing special about those variable names. Instead I'm interested in cases satisfying the following conditions:
Mathematicians actually working with these objects commonly represent the things they are adding or multiplying or whatever (in general, performing algebraic manipulations on) by pictures.
The pictures used to represent these objects are actually helpful for understanding what the objects are.
It's okay if it's not adding or multiplying specifically we're doing, as long as we're manipulating the pictures in ways traditionally reserved for numbers or variables. For example, the things represented by pictures could be elements of some algebraic object (group, ring, etc.)
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